Research Article
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Year 2020, Volume: 49 Issue: 1, 389 - 398, 06.02.2020
https://doi.org/10.15672/hujms.568258

Abstract

References

  • [1] H. Arora and R. Karan, What is the probability an automorphism fixes a group element?, Comm. Algebra, 45(3), 1141–1150, 2017.
  • [2] A.K. Das and R.K. Nath, On generalized relative commutativity degree of a finite group, Int. Electron. J. Algebra, 7, 140–151, 2010.
  • [3] P. Dutta and R.K. Nath, Autocommuting probabilty of a finite group, Comm. Algebra, 46 (3), 961–969, 2018.
  • [4] P. Dutta and R.K. Nath, On generalized autocommutativity degree of finite groups, Hacet. J. Math. Stat. 48 (2), 472–478, 2019.
  • [5] P. Hall, The classification of prime power groups, J. Reine Angew. Math. 182, 130– 141, 1940.
  • [6] P.V. Hegarty, The absolute centre of a group, J. Algebra, 169 (3), 929–935, 1994.
  • [7] C.J. Hillar and D.L. Rhea, Automorphism of finite abelian groups, Amer. Math. Monthly, 114 (10), 917–923, 2007.
  • [8] M.R.R. Moghaddam, M.J. Sadeghifard and M. Eshrati, Some properties of autoisoclinism of groups, Fifth International group theory conference, Islamic Azad University, Mashhad, Iran, 13-15 March 2013.
  • [9] M.R.R. Moghaddam, F. Saeedi and E. Khamseh, The probability of an automorphism fixing a subgroup element of a finite group, Asian-Eur. J. Math. 4 (2), 301–308, 2011.
  • [10] R.K. Nath and A.K. Das, On a lower bound of commutativity degree, Rend. Circ. Mat. Palermo, 59 (1), 137–142, 2010.
  • [11] R.K. Nath and M.K. Yadav, Some results on relative commutativity degree, Rend. Circ. Mat. Palermo, 64 (2), 229–239, 2015.
  • [12] M.R. Rismanchian and Z. Sepehrizadeh, Autoisoclinism classes and autocommutativity degrees of finite groups, Hacet. J. Math. Stat. 44 (4), 893–899, 2015.
  • [13] G.J. Sherman, What is the probability an automorphism fixes a group element?, Amer. Math. Monthly, 82, 261–264, 1975.

Generalized autocommuting probability of a finite group relative to its subgroups

Year 2020, Volume: 49 Issue: 1, 389 - 398, 06.02.2020
https://doi.org/10.15672/hujms.568258

Abstract

Let $H \subseteq K$ be two subgroups of a finite group $G$  and $\mathrm{Aut}(K)$ the automorphism group of  $K$. In this paper, we consider the generalized autocommuting probability of $G$ relative to its subgroups $H$ and $K$, denoted by  ${Pr}_g(H,\mathrm{Aut}(K))$, which is the probability  that the autocommutator of a randomly chosen pair of elements, one from $H$ and the other from $\mathrm{Aut}(K)$, is equal to a given element $g \in K$. We study several properties as well as obtain several computing formulae of  this probability. As applications of the computing formulae, we also obtain several  bounds for ${Pr}_g(H,\mathrm{Aut}(K))$ and characterizations of some finite groups through ${Pr}_g(H,\mathrm{Aut}(K))$.

References

  • [1] H. Arora and R. Karan, What is the probability an automorphism fixes a group element?, Comm. Algebra, 45(3), 1141–1150, 2017.
  • [2] A.K. Das and R.K. Nath, On generalized relative commutativity degree of a finite group, Int. Electron. J. Algebra, 7, 140–151, 2010.
  • [3] P. Dutta and R.K. Nath, Autocommuting probabilty of a finite group, Comm. Algebra, 46 (3), 961–969, 2018.
  • [4] P. Dutta and R.K. Nath, On generalized autocommutativity degree of finite groups, Hacet. J. Math. Stat. 48 (2), 472–478, 2019.
  • [5] P. Hall, The classification of prime power groups, J. Reine Angew. Math. 182, 130– 141, 1940.
  • [6] P.V. Hegarty, The absolute centre of a group, J. Algebra, 169 (3), 929–935, 1994.
  • [7] C.J. Hillar and D.L. Rhea, Automorphism of finite abelian groups, Amer. Math. Monthly, 114 (10), 917–923, 2007.
  • [8] M.R.R. Moghaddam, M.J. Sadeghifard and M. Eshrati, Some properties of autoisoclinism of groups, Fifth International group theory conference, Islamic Azad University, Mashhad, Iran, 13-15 March 2013.
  • [9] M.R.R. Moghaddam, F. Saeedi and E. Khamseh, The probability of an automorphism fixing a subgroup element of a finite group, Asian-Eur. J. Math. 4 (2), 301–308, 2011.
  • [10] R.K. Nath and A.K. Das, On a lower bound of commutativity degree, Rend. Circ. Mat. Palermo, 59 (1), 137–142, 2010.
  • [11] R.K. Nath and M.K. Yadav, Some results on relative commutativity degree, Rend. Circ. Mat. Palermo, 64 (2), 229–239, 2015.
  • [12] M.R. Rismanchian and Z. Sepehrizadeh, Autoisoclinism classes and autocommutativity degrees of finite groups, Hacet. J. Math. Stat. 44 (4), 893–899, 2015.
  • [13] G.J. Sherman, What is the probability an automorphism fixes a group element?, Amer. Math. Monthly, 82, 261–264, 1975.
There are 13 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Parama Dutta This is me 0000-0002-6984-9817

Rajat Nath 0000-0003-4766-6523

Publication Date February 6, 2020
Published in Issue Year 2020 Volume: 49 Issue: 1

Cite

APA Dutta, P., & Nath, R. (2020). Generalized autocommuting probability of a finite group relative to its subgroups. Hacettepe Journal of Mathematics and Statistics, 49(1), 389-398. https://doi.org/10.15672/hujms.568258
AMA Dutta P, Nath R. Generalized autocommuting probability of a finite group relative to its subgroups. Hacettepe Journal of Mathematics and Statistics. February 2020;49(1):389-398. doi:10.15672/hujms.568258
Chicago Dutta, Parama, and Rajat Nath. “Generalized Autocommuting Probability of a Finite Group Relative to Its Subgroups”. Hacettepe Journal of Mathematics and Statistics 49, no. 1 (February 2020): 389-98. https://doi.org/10.15672/hujms.568258.
EndNote Dutta P, Nath R (February 1, 2020) Generalized autocommuting probability of a finite group relative to its subgroups. Hacettepe Journal of Mathematics and Statistics 49 1 389–398.
IEEE P. Dutta and R. Nath, “Generalized autocommuting probability of a finite group relative to its subgroups”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 1, pp. 389–398, 2020, doi: 10.15672/hujms.568258.
ISNAD Dutta, Parama - Nath, Rajat. “Generalized Autocommuting Probability of a Finite Group Relative to Its Subgroups”. Hacettepe Journal of Mathematics and Statistics 49/1 (February 2020), 389-398. https://doi.org/10.15672/hujms.568258.
JAMA Dutta P, Nath R. Generalized autocommuting probability of a finite group relative to its subgroups. Hacettepe Journal of Mathematics and Statistics. 2020;49:389–398.
MLA Dutta, Parama and Rajat Nath. “Generalized Autocommuting Probability of a Finite Group Relative to Its Subgroups”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 1, 2020, pp. 389-98, doi:10.15672/hujms.568258.
Vancouver Dutta P, Nath R. Generalized autocommuting probability of a finite group relative to its subgroups. Hacettepe Journal of Mathematics and Statistics. 2020;49(1):389-98.